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    Re: Unexpected USNO height correction precepts
    From: Antoine Couëtte
    Date: 2018 Sep 23, 09:55 -0700

    Hello Paul,

    Replying here to your last 2 posts on this subject.

    1 - RE: Unexpected-USNO-height-correction-precepts-Hirose-sep-2018-g42759

    In this post you stated:

     "A note near the bottom of that page says, "The data for Venus has not been corrected for phase; under certain conditions this could displace the center of light up to 0.3 arcminutes." But Antoine says his example has .7' error if phase is ignored.

    Actually, yes I am observing a 0.7' error in my example since not only phase error is ignored by the USNO, but also the USNO site also [wrongly] includes the Planet semi-diameter in its published [overall height Correction] "Sum". It is totally irrelevant since Venus was observed as a very faint pinpoint bright spot in the sextants. Hence we did observe the center of light itself, and certainly not the lower limb as is implied by the USNO "Sums" published values which - again and for some very strange reason - do include Planets' SD's.

    1.a - Phase correction general Formula

    I am using our most regretted George Huxtable's recommended formula published in NavList [11961] on Sep 18th, 2010.

    Let SD be the angular semi-diameter seen from Earth center, "φ" the phase angle, then the phase correction "λ" is computed as follows :

    λ = [ 8 / (3π) ] * sin² (φ/2) * SD

    1.b - Numerical Application

    As per your request here-under, I am giving here-after extra digits to 0.01' to be sure that round off errors do not "contaminate" this numerical example, although extra-digits are not really necessary in this specific example.

    - Venus semi-diameter : 0.42'

    As a numerical application, with φ = 139° as stated in your second post referenced here-under, then λ = [ 8 / 9.425... ] * sin² (139°/2) * 0.42' = 0.31' .

    As earlier stated, the Phase correction "λ" is reckoned from Venus Center towards the Sun Center.

    λ  can be computed in any reference system : Ecliptic as [δλδβ], or Equator as [δRA, δDEC] , or Horizontal as [δAz, δALT] . This geocentric angular correction is simply reckoned from the Planet Center towards the Sun Center and it has the angular magnitude specified here-above.

    I am actually computing λ in the plan containing Sun Center - Planet Center - Earth Center since λ then entirely shows up as a unique correction to the angular variable measured in this specific plan.

    Due to the very special configuration of both Venus and the Sun being in almost the very same Azimuth in this example, λ also shows up here as a simple and unique heigh correction (no Azimuth correction) from Venus center towards the Sun Center which is higher. Accordingly the center of Venus light is exactly above the center of Venus by a quantity equal to 0.31' as earlier computed.

    To recap, the USNO rationale seems to imply: "you are observing the Planet Lower Limb height, and I am computing you an intercept from that observed part of the Planet".

    Instead of this USNO precept: "I am observing instead the height of the Planet Center of Light and I am computing an Intercept from this observed part of the Planet".

    Since in this [extreme] case, the respective heights of both points differ by exactly " SD + λ " = 0.42' + 0.31' = 0.73' , then no surprise at all to see that - as you earlier stated -  "[t]his example has .7' error if phase is ignored" . Hint ! Remember that I am also ignoring the Planet's SD while the USNO takes in account planet SD in its "Sum" height correction.

    As a full summary : Venus Phase effect should desirably be taken in account in the USNO published height reduction process. And for all height correction purposes, the USNO should set the Planets SD's equal to 0. This does not seem to be the case as the currently published "Sums" do include the Planets SD's, hence potentially misleading unfamiliar users.

    Paul, most of your queries should have been covered in the first part of this current post. Hence, in the following part I am simpling comparing our results.

    2 - RE: Unexpected-USNO-height-correction-precepts-Hirose-sep-2018-g42763

    1977 Mar 18 at 00:02:17.0 UT
    DR Position S 27°35.0’  E 144°53.7  Height of Eye 15’
    P = 1010 Mb T = +10°C
    Height observed in Sextant: 18°03.7’ Index Error 0.0’
    Nautical Almanac Dip Correction: -3.8’
    For a lower limb Venus observation Lunar4 says:      Antoine's results
          3.79′ dip (calculated by Lunar4)               -3.78' dip correction
      17°59.91′ apparent lower limb altitude              17°59.92' apparent Center of Light Altitude
          2.94′ refraction                               -3.02'     refraction correction
         -0.42′ unrefracted semidiameter                  0.42'     unrefracted SD. I am taking SD=0 since my only reference remains Center of Light  
      17°57.39′ unrefracted altitude of center            17°56.90' unrefracted altitude of Center of Light                                                  
      17°57.67′ predicted altitude                       +0.42'     Parallax Correction                                                      
                                                          17°57.32' Geocentric OBSERVED  height of Center of Light
                                                          17°58.45' Geocentric PREDICTED height of Center of Light
      -0°00.28′ intercept                                -1.13'     Intercept
      59°49.46′ predicted azimuth                        59°49.42'   predicted azimuth
    +0m47.52s delta T                                  +0m47.52s delta T (I intentionally used your own value)
    Discussion :
    0.85' of difference between our intercepts. Allowing for differences in Refraction, the difference between our intercepts boils down to 0.77' . This is to be compared to the  " SD + λ " = 0.42' + 0.31' = 0.73'  value listed hereabove.
    This difference in our intercepts can then be fully explained by the fact that you are following here-above the full USNO precepts - i.e. ignoring Phase Correction and assuming that the Lower Limb was actually observed - while on the other hand I am taking in effect the Phase correction and I am also considering that I am observing a pin-point light spot with no appreciable apparent semi-diameter in full daylight and which is in fact "far away" (at some 0.7' in height) from the Planet Lower Limb.
    This should cover it all.
    Best Friendly Regards, and thank you again for your feedback.
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